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mfb said:If there is no matter falling in, we would just note the gravitational lensing effect on starlight passing close to it, most notably in Gaia data.
The heliopause and termination shock would move further from the sun in the direction of the black hole.
Why in that direction?
There would be some matter falling in. The sparse interstellar gas of Local Bubble.
But if you have a black hole moving at high relative speed through sparse interstellar gas, what kind of disturbance is created? The gas accelerates on approach to the hole - then falls past event horizon with most of its kinetic energy.